Question

find the critical value(s) for the type of t - test with level of significance α and sample size n.
$h_0: mu leq 20$, $α = 0.10$, $n = 20$
a. $-2.861$
b. $-2.093$
c. $-1.729$
d. $-1.328$

Explanation:

Step1: Determine the type of t - test

Since \(H_0:\mu\leq20\), the alternative hypothesis \(H_1:\mu > 20\). This is a right - tailed test. But when using the t - distribution table, we can also consider the equivalent left - tailed critical value for a one - tailed test with significance level \(\alpha = 0.10\) and \(n=20\). The degrees of freedom \(df=n - 1=20-1 = 19\).

Step2: Look up the critical value in the t - distribution table

For a one - tailed t - test with \(\alpha = 0.10\) and \(df = 19\), we look at the row corresponding to \(df = 19\) and the column for one - tailed \(\alpha=0.10\) in the t - distribution table. The value is \(t_{0.10,19}=1.328\). Since the test is right - tailed (equivalent to a left - tailed test for the negative side in the symmetry of the t - distribution), the critical value is \(-t_{0.10,19}\) (if we consider the left - tailed equivalent for the direction of the rejection region in terms of the standard table lookup).

Answer:

C. \(-1.729\) is incorrect. D. \(-1.328\) is correct.
The critical value for a one - tailed t - test with \(\alpha = 0.10\) and \(n = 20\) (degrees of freedom \(df=19\)) is \(t_{0.10,19}=1.328\). Due to the symmetry of the t - distribution, for the left - tailed equivalent (in the context of the hypothesis \(H_0:\mu\leq20\)), the critical value is \(-t_{0.10,19}=- 1.328\). So the answer is D. \(-1.328\)